OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING

Abstract

A mathematical framework for Continuous Time Finance based on operator algebraic
methods oers a new direct and entirely constructive perspective on the field. It also
leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations.

This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can eectively use block-factorization algorithms. Finally, we discuss
the method of dynamic conditioning that allows one to dynamically correlate over possibly
even hundreds of processes in a numerically noiseless framework while preserving marginal
distributions.